Q: What is the margin of error for a 95% confidence interval for a sample mean, given $$\sigma = 10$$, $$n = 100$$?

$$E = z^* \frac{\sigma}{\sqrt{n}} = 1.96 \times \frac{10}{10} = 1.96$$

Q: Given a sample mean $$\bar{x} = 75$$, population standard deviation $$\sigma = 8$$, sample size $$n = 64$$, construct a 95% confidence interval for the population mean.

$$75 \pm 1.96 \times \frac{8}{8} = 75 \pm 1.96$$

Question 1

A researcher collects a random sample of 100 patients and finds that 60% of them respond positively to a new medication. What is the best estimate of the population proportion of patients who would respond positively to the medication?

Accepted Answer

The sample proportion, $$\hat{p} = 0.60$$

Question 2

A sample of 36 students has a mean test score of 80 with a population standard deviation of 12. What is the z-score for a sample mean of 83?

Accepted Answer

$$z = \frac{83 - 80}{12 / \sqrt{36}} = \frac{3}{2} = 1.5$$

Question 3

If the z-score for a sample mean is 2.0, what does this indicate?

Accepted Answer

The sample mean is 2 standard deviations above the mean of the distribution of sample means.

Question 4

You look up a z-score of 1.25 in the z-table and find the area to the left is 0.8944. What is the probability of getting a value greater than this z-score?

Accepted Answer

$$P(X \u003e z) = 1 - 0.8944 = 0.1056$$

Question 5

How do you determine the probability for a two-tailed test with a z-score of $$\pm 2.0$$?

Accepted Answer

Find the area to the left of $$z = -2.0$$ and the area to the right of $$z = 2.0$$, then add them together.

Question 6

Which formula is used to calculate the sample proportion?

Accepted Answer

$$\hat{p} = \frac{x}{n}$$

Question 7

A poll of 400 voters finds that 220 support a new law. What is the sample proportion of voters who support the law?

Accepted Answer

$$\hat{p} = \frac{220}{400} = 0.55$$

Question 8

What is the margin of error for a 95% confidence interval for a sample mean, given $$\sigma = 10$$, $$n = 100$$?

Accepted Answer

$$E = z^* \frac{\sigma}{\sqrt{n}} = 1.96 	imes \frac{10}{10} = 1.96$$

Question 9

A researcher wants a margin of error of no more than 3% for a 95% confidence interval for a proportion. What sample size is needed if the estimated proportion is 0.5?

Accepted Answer

$$n = \left( \frac{z^*}{E} \$$right)^2$$ p(1-p) = \left( \frac{1.96}{0.03} \$$right)^2$$ 	imes 0.5 	imes 0.5 = 1067$$

Question 10

Given a sample mean $$\bar{x} = 75$$, population standard deviation $$\sigma = 8$$, sample size $$n = 64$$, construct a 95% confidence interval for the population mean.

Accepted Answer

$$75 \pm 1.96 	imes \frac{8}{8} = 75 \pm 1.96$$

Question 11

Which of the following is a correct null hypothesis for testing whether a new drug changes the average recovery time from an illness (population mean is 10 days)?

Accepted Answer

$$H_0$$: \mu = 10$$

Question 12

If you reject the null hypothesis in a hypothesis test, what does this mean?

Accepted Answer

There is sufficient evidence to support the alternative hypothesis.

Question 13

A p-value of 0.03 is obtained in a test. Is this result significant at the 0.05 level?

Accepted Answer

Yes, because $$0.03 \u003c 0.05$$

Question 14

A z-score of 2.5 is calculated for a sample mean. Using a standard normal table, the area to the left is 0.9938. What is the probability of observing a value greater than this z-score?

Accepted Answer

$$1 - 0.9938 = 0.0062$$

Question 15

Which of the following best describes a Type I error?

Accepted Answer

Rejecting the null hypothesis when it is actually true.

Question 16

Which of the following is an example of a Type II error in a medical trial?

Accepted Answer

Concluding a new drug is not effective when it actually is.

Question 17

A study finds a p-value of 0.008. At which significance levels is this result statistically significant?

Accepted Answer

At both $$\alpha = 0.05$$ and $$\alpha = 0.01$$

Question 18

A sample of 49 patients has a mean blood pressure of 120 mmHg. The population standard deviation is 14 mmHg. What is the 95% confidence interval for the mean blood pressure?

Accepted Answer

$$120 \pm 1.96 	imes \frac{14}{7} = 120 \pm 3.92$$

Question 19

A company wants to estimate the proportion of defective products with a margin of error of 2% at 95% confidence. If the estimated proportion is 0.1, what sample size is required?

Accepted Answer

$$n = \left( \frac{1.96}{0.02} \$$right)^2$$ 	imes 0.1 	imes 0.9 = 864$$

Statistics Test 3 Study Guide: Z-Scores, Confidence Intervals, Hypothesis Testing, and Proportions

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